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Rough Burger-like SPDEs

Nannan Li, Xing Gao

TL;DR

The work addresses pathwise well-posedness of Burgers-type SPDEs driven by additive space-time white noise in one dimension. It extends rough path methods to the subcritical regime $\alpha\in(0,\tfrac{1}{2})$ by developing refined estimates for controlled rough paths and rough integrals against heat kernels. Key contributions include a new upper bound for compositions with regular functions, refined scaling arguments for rough integrals, and a global pathwise well-posedness theory for mild solutions under suitable boundedness assumptions. Collectively, these advances broaden the class of Burgers-type SPDEs that can be treated pathwise with rough path techniques and lay groundwork for deeper analyses of highly singular stochastic PDEs.

Abstract

We study a class of nonlinear Burgers-type stochastic partial differential equations driven by additive space-time white noise in one spatial dimension. Building on the rough path framework initiated by Hairer, which provides a pathwise solution theory under spatial regularity $α\in(\frac{1}{3}, \frac{1}{2})$, we extend this approach to the full subcritical regime $α\in(0, \frac{1}{2})$. Our main contribution is the establishment of pathwise existence and uniqueness of mild (equivalently, weak) solutions when the spatial regularity of the solution lies strictly below the classical rough path threshold. This is achieved through refined estimates for controlled rough paths, including a new upper bound for compositions with smooth functions and a scaling analysis for rough integrals against heat kernels. In particular, we extend and sharpen key analytic estimates originating from Hairer's work, incorporating refined scaling arguments that are effective in the low-regularity regime. As a result, our framework significantly enlarges the class of Burgers-type SPDEs that can be treated pathwise using rough path techniques.

Rough Burger-like SPDEs

TL;DR

The work addresses pathwise well-posedness of Burgers-type SPDEs driven by additive space-time white noise in one dimension. It extends rough path methods to the subcritical regime by developing refined estimates for controlled rough paths and rough integrals against heat kernels. Key contributions include a new upper bound for compositions with regular functions, refined scaling arguments for rough integrals, and a global pathwise well-posedness theory for mild solutions under suitable boundedness assumptions. Collectively, these advances broaden the class of Burgers-type SPDEs that can be treated pathwise with rough path techniques and lay groundwork for deeper analyses of highly singular stochastic PDEs.

Abstract

We study a class of nonlinear Burgers-type stochastic partial differential equations driven by additive space-time white noise in one spatial dimension. Building on the rough path framework initiated by Hairer, which provides a pathwise solution theory under spatial regularity , we extend this approach to the full subcritical regime . Our main contribution is the establishment of pathwise existence and uniqueness of mild (equivalently, weak) solutions when the spatial regularity of the solution lies strictly below the classical rough path threshold. This is achieved through refined estimates for controlled rough paths, including a new upper bound for compositions with smooth functions and a scaling analysis for rough integrals against heat kernels. In particular, we extend and sharpen key analytic estimates originating from Hairer's work, incorporating refined scaling arguments that are effective in the low-regularity regime. As a result, our framework significantly enlarges the class of Burgers-type SPDEs that can be treated pathwise using rough path techniques.
Paper Structure (13 sections, 8 theorems, 114 equations)

This paper contains 13 sections, 8 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. Burgers-like SPDEs
  3. Rough paths
  4. Comparison and outline
  5. An upper bound of the composition with regular functions
  6. Rough paths and controlled rough paths
  7. Composition with regular functions
  8. Rough integrals and scaled functions
  9. Rough integrals
  10. Integration with scaled functions
  11. Well-posedness
  12. Gaussian rough paths and definition of solutions
  13. Existence and uniqueness

Key Result

Proposition 2.6

Let $\alpha \in(0,1]$, ${\bf X}\in {\mathcal{D}}^{\alpha}_{w}([0, T]^2, \mathbb{R}^d)$, ${\bf Y} = (Y^0, \ldots, Y^{N-1})\in \mathcal{C}_{{\bf X}}^{\alpha}([0, T], \mathbb{R}^m)$ and $\varphi({\bf Y}) = (\varphi(Y)^0, \ldots, \varphi(Y)^{N-1})\in \mathcal{C}_{{\bf X}}^{\alpha}([0, T], \mathbb{R}^n)$ where $C_{\alpha, T}\in \mathbb{R}$ and $l, k \in \mathbb{R}_{>0}$.

Theorems & Definitions (24)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Remark 2.4
  • Definition 2.5
  • Proposition 2.6
  • proof
  • Remark 2.7
  • Definition 3.1
  • Lemma 3.2
  • ...and 14 more